Subgroup growth in some pro-$p$ groups

Abstract:

For a group $G$ let $a_{n}(G)$ be the number of subgroups of index $n$ and let $b_{n}(G)$ be the number of normal subgroups of index $n$. We show that $a_{p^{k}}(SL_{2}^{1}(\mathbb {F}_{p}[[t]])) \le p^{k(k+5)/2}$ for $p>2$. If $\Lambda =\mathbb {F}_{p}[[t]]$ and $p$ does not divide $d$ or if $\Lambda =\mathbb {Z}_{p}$ and $p \ne 2$ or $d \ne 2$, we show that for all $k$ sufficiently large $b_{p^{k}}(SL_{d}^{1}(\Lambda ))=b_{p^{k+d^{2}-1}}(SL_{d}^{1}(\Lambda ))$. On the other hand if $\Lambda =\mathbb {F}_{p}[[t]]$ and $p$ divides $d$, then $b_{n}(SL_{d}^{1}(\Lambda ))$ is not even bounded as a function of $n$.